The chain is bouncing back quickly from a prescription dispute, expanding internationally, and should benefit as an aging population heads to drugstores for basic health-care needs.

Not only do drugstores seem to be popping up on every intersection, they are also inserting themselves into more corners of our lives. Once a place to pick up a prescription, a tube of toothpaste and a pack of gum, they have morphed into grocery store, photo printing service, and, in a few extreme cases, even beer bars.

For their next act, drugstores are morphing from stores that sell medicine (and India Pale Ale) into health-care centers that also see patients. For Walgreen (ticker: WAG), the drug and beauty retailer,...